Properties

Label 8037.2.a.i.1.3
Level $8037$
Weight $2$
Character 8037.1
Self dual yes
Analytic conductor $64.176$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8037,2,Mod(1,8037)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8037, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8037.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8037.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.1757681045\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.5476681.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 11x^{4} + 19x^{3} + 32x^{2} - 44x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2679)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.41126\) of defining polynomial
Character \(\chi\) \(=\) 8037.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.554958 q^{2} -1.69202 q^{4} -2.17107 q^{5} +2.41126 q^{7} +2.04892 q^{8} +O(q^{10})\) \(q-0.554958 q^{2} -1.69202 q^{4} -2.17107 q^{5} +2.41126 q^{7} +2.04892 q^{8} +1.20486 q^{10} +1.55496 q^{11} -1.12216 q^{13} -1.33815 q^{14} +2.24698 q^{16} +5.71408 q^{17} -1.00000 q^{19} +3.67350 q^{20} -0.862937 q^{22} -1.08270 q^{23} -0.286438 q^{25} +0.622750 q^{26} -4.07990 q^{28} +0.609323 q^{29} -2.39902 q^{31} -5.34481 q^{32} -3.17107 q^{34} -5.23503 q^{35} -7.29892 q^{37} +0.554958 q^{38} -4.44835 q^{40} -6.12134 q^{41} -5.93076 q^{43} -2.63102 q^{44} +0.600852 q^{46} +1.00000 q^{47} -1.18582 q^{49} +0.158961 q^{50} +1.89871 q^{52} -8.54484 q^{53} -3.37593 q^{55} +4.94047 q^{56} -0.338149 q^{58} +6.46496 q^{59} +5.93768 q^{61} +1.33136 q^{62} -1.52781 q^{64} +2.43629 q^{65} -0.887061 q^{67} -9.66834 q^{68} +2.90522 q^{70} +12.6724 q^{71} -8.92007 q^{73} +4.05060 q^{74} +1.69202 q^{76} +3.74941 q^{77} +14.7821 q^{79} -4.87836 q^{80} +3.39709 q^{82} -3.18414 q^{83} -12.4057 q^{85} +3.29132 q^{86} +3.18598 q^{88} +13.9012 q^{89} -2.70581 q^{91} +1.83195 q^{92} -0.554958 q^{94} +2.17107 q^{95} +7.63568 q^{97} +0.658081 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} + 4 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{2} + 4 q^{5} + 2 q^{7} - 6 q^{8} + 2 q^{10} + 10 q^{11} - 8 q^{13} + q^{14} + 4 q^{16} + 3 q^{17} - 6 q^{19} - 7 q^{20} - 16 q^{22} + 10 q^{26} - 7 q^{28} - 2 q^{31} + 14 q^{32} - 2 q^{34} - 7 q^{35} - 9 q^{37} + 4 q^{38} - 4 q^{40} - 18 q^{41} + 9 q^{43} + 14 q^{44} - 7 q^{46} + 6 q^{47} - 16 q^{49} + 7 q^{50} - 21 q^{52} - 20 q^{53} + 2 q^{55} + 5 q^{56} + 7 q^{58} - 19 q^{59} + 13 q^{62} - 22 q^{64} + 22 q^{65} - 11 q^{67} - 14 q^{70} - 4 q^{71} - 7 q^{73} + 27 q^{74} + q^{77} - 16 q^{79} + 5 q^{80} - 2 q^{82} - 7 q^{83} - 25 q^{85} + q^{86} - 10 q^{88} + 33 q^{89} - 4 q^{91} + 21 q^{92} - 4 q^{94} - 4 q^{95} - 34 q^{97} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.554958 −0.392415 −0.196207 0.980562i \(-0.562863\pi\)
−0.196207 + 0.980562i \(0.562863\pi\)
\(3\) 0 0
\(4\) −1.69202 −0.846011
\(5\) −2.17107 −0.970934 −0.485467 0.874255i \(-0.661350\pi\)
−0.485467 + 0.874255i \(0.661350\pi\)
\(6\) 0 0
\(7\) 2.41126 0.911371 0.455685 0.890141i \(-0.349394\pi\)
0.455685 + 0.890141i \(0.349394\pi\)
\(8\) 2.04892 0.724402
\(9\) 0 0
\(10\) 1.20486 0.381009
\(11\) 1.55496 0.468838 0.234419 0.972136i \(-0.424681\pi\)
0.234419 + 0.972136i \(0.424681\pi\)
\(12\) 0 0
\(13\) −1.12216 −0.311230 −0.155615 0.987818i \(-0.549736\pi\)
−0.155615 + 0.987818i \(0.549736\pi\)
\(14\) −1.33815 −0.357635
\(15\) 0 0
\(16\) 2.24698 0.561745
\(17\) 5.71408 1.38587 0.692934 0.721001i \(-0.256318\pi\)
0.692934 + 0.721001i \(0.256318\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 3.67350 0.821420
\(21\) 0 0
\(22\) −0.862937 −0.183979
\(23\) −1.08270 −0.225758 −0.112879 0.993609i \(-0.536007\pi\)
−0.112879 + 0.993609i \(0.536007\pi\)
\(24\) 0 0
\(25\) −0.286438 −0.0572875
\(26\) 0.622750 0.122131
\(27\) 0 0
\(28\) −4.07990 −0.771030
\(29\) 0.609323 0.113148 0.0565742 0.998398i \(-0.481982\pi\)
0.0565742 + 0.998398i \(0.481982\pi\)
\(30\) 0 0
\(31\) −2.39902 −0.430877 −0.215438 0.976517i \(-0.569118\pi\)
−0.215438 + 0.976517i \(0.569118\pi\)
\(32\) −5.34481 −0.944839
\(33\) 0 0
\(34\) −3.17107 −0.543835
\(35\) −5.23503 −0.884881
\(36\) 0 0
\(37\) −7.29892 −1.19994 −0.599968 0.800024i \(-0.704820\pi\)
−0.599968 + 0.800024i \(0.704820\pi\)
\(38\) 0.554958 0.0900261
\(39\) 0 0
\(40\) −4.44835 −0.703346
\(41\) −6.12134 −0.955993 −0.477996 0.878362i \(-0.658637\pi\)
−0.477996 + 0.878362i \(0.658637\pi\)
\(42\) 0 0
\(43\) −5.93076 −0.904432 −0.452216 0.891908i \(-0.649366\pi\)
−0.452216 + 0.891908i \(0.649366\pi\)
\(44\) −2.63102 −0.396642
\(45\) 0 0
\(46\) 0.600852 0.0885908
\(47\) 1.00000 0.145865
\(48\) 0 0
\(49\) −1.18582 −0.169403
\(50\) 0.158961 0.0224805
\(51\) 0 0
\(52\) 1.89871 0.263304
\(53\) −8.54484 −1.17372 −0.586862 0.809687i \(-0.699637\pi\)
−0.586862 + 0.809687i \(0.699637\pi\)
\(54\) 0 0
\(55\) −3.37593 −0.455210
\(56\) 4.94047 0.660199
\(57\) 0 0
\(58\) −0.338149 −0.0444011
\(59\) 6.46496 0.841666 0.420833 0.907138i \(-0.361738\pi\)
0.420833 + 0.907138i \(0.361738\pi\)
\(60\) 0 0
\(61\) 5.93768 0.760242 0.380121 0.924937i \(-0.375882\pi\)
0.380121 + 0.924937i \(0.375882\pi\)
\(62\) 1.33136 0.169082
\(63\) 0 0
\(64\) −1.52781 −0.190976
\(65\) 2.43629 0.302184
\(66\) 0 0
\(67\) −0.887061 −0.108372 −0.0541859 0.998531i \(-0.517256\pi\)
−0.0541859 + 0.998531i \(0.517256\pi\)
\(68\) −9.66834 −1.17246
\(69\) 0 0
\(70\) 2.90522 0.347240
\(71\) 12.6724 1.50394 0.751970 0.659198i \(-0.229104\pi\)
0.751970 + 0.659198i \(0.229104\pi\)
\(72\) 0 0
\(73\) −8.92007 −1.04402 −0.522008 0.852941i \(-0.674817\pi\)
−0.522008 + 0.852941i \(0.674817\pi\)
\(74\) 4.05060 0.470872
\(75\) 0 0
\(76\) 1.69202 0.194088
\(77\) 3.74941 0.427285
\(78\) 0 0
\(79\) 14.7821 1.66312 0.831560 0.555435i \(-0.187448\pi\)
0.831560 + 0.555435i \(0.187448\pi\)
\(80\) −4.87836 −0.545417
\(81\) 0 0
\(82\) 3.39709 0.375146
\(83\) −3.18414 −0.349505 −0.174753 0.984612i \(-0.555913\pi\)
−0.174753 + 0.984612i \(0.555913\pi\)
\(84\) 0 0
\(85\) −12.4057 −1.34559
\(86\) 3.29132 0.354913
\(87\) 0 0
\(88\) 3.18598 0.339627
\(89\) 13.9012 1.47353 0.736765 0.676149i \(-0.236353\pi\)
0.736765 + 0.676149i \(0.236353\pi\)
\(90\) 0 0
\(91\) −2.70581 −0.283646
\(92\) 1.83195 0.190994
\(93\) 0 0
\(94\) −0.554958 −0.0572396
\(95\) 2.17107 0.222747
\(96\) 0 0
\(97\) 7.63568 0.775286 0.387643 0.921810i \(-0.373289\pi\)
0.387643 + 0.921810i \(0.373289\pi\)
\(98\) 0.658081 0.0664763
\(99\) 0 0
\(100\) 0.484659 0.0484659
\(101\) 14.4415 1.43698 0.718492 0.695535i \(-0.244833\pi\)
0.718492 + 0.695535i \(0.244833\pi\)
\(102\) 0 0
\(103\) 4.49612 0.443016 0.221508 0.975159i \(-0.428902\pi\)
0.221508 + 0.975159i \(0.428902\pi\)
\(104\) −2.29921 −0.225456
\(105\) 0 0
\(106\) 4.74203 0.460587
\(107\) 0.462142 0.0446769 0.0223385 0.999750i \(-0.492889\pi\)
0.0223385 + 0.999750i \(0.492889\pi\)
\(108\) 0 0
\(109\) 0.337009 0.0322796 0.0161398 0.999870i \(-0.494862\pi\)
0.0161398 + 0.999870i \(0.494862\pi\)
\(110\) 1.87350 0.178631
\(111\) 0 0
\(112\) 5.41805 0.511958
\(113\) 2.93103 0.275728 0.137864 0.990451i \(-0.455976\pi\)
0.137864 + 0.990451i \(0.455976\pi\)
\(114\) 0 0
\(115\) 2.35062 0.219196
\(116\) −1.03099 −0.0957248
\(117\) 0 0
\(118\) −3.58778 −0.330282
\(119\) 13.7781 1.26304
\(120\) 0 0
\(121\) −8.58211 −0.780191
\(122\) −3.29516 −0.298330
\(123\) 0 0
\(124\) 4.05919 0.364526
\(125\) 11.4772 1.02656
\(126\) 0 0
\(127\) −16.4566 −1.46028 −0.730142 0.683295i \(-0.760546\pi\)
−0.730142 + 0.683295i \(0.760546\pi\)
\(128\) 11.5375 1.01978
\(129\) 0 0
\(130\) −1.35204 −0.118581
\(131\) 13.5020 1.17968 0.589839 0.807521i \(-0.299191\pi\)
0.589839 + 0.807521i \(0.299191\pi\)
\(132\) 0 0
\(133\) −2.41126 −0.209083
\(134\) 0.492281 0.0425266
\(135\) 0 0
\(136\) 11.7077 1.00392
\(137\) −7.13332 −0.609440 −0.304720 0.952442i \(-0.598563\pi\)
−0.304720 + 0.952442i \(0.598563\pi\)
\(138\) 0 0
\(139\) 9.04370 0.767077 0.383538 0.923525i \(-0.374705\pi\)
0.383538 + 0.923525i \(0.374705\pi\)
\(140\) 8.85778 0.748619
\(141\) 0 0
\(142\) −7.03266 −0.590168
\(143\) −1.74491 −0.145916
\(144\) 0 0
\(145\) −1.32289 −0.109860
\(146\) 4.95027 0.409687
\(147\) 0 0
\(148\) 12.3499 1.01516
\(149\) 7.19506 0.589443 0.294721 0.955583i \(-0.404773\pi\)
0.294721 + 0.955583i \(0.404773\pi\)
\(150\) 0 0
\(151\) −13.5275 −1.10085 −0.550424 0.834885i \(-0.685534\pi\)
−0.550424 + 0.834885i \(0.685534\pi\)
\(152\) −2.04892 −0.166189
\(153\) 0 0
\(154\) −2.08077 −0.167673
\(155\) 5.20845 0.418353
\(156\) 0 0
\(157\) −6.02659 −0.480975 −0.240487 0.970652i \(-0.577307\pi\)
−0.240487 + 0.970652i \(0.577307\pi\)
\(158\) −8.20346 −0.652633
\(159\) 0 0
\(160\) 11.6040 0.917376
\(161\) −2.61067 −0.205749
\(162\) 0 0
\(163\) −22.0480 −1.72693 −0.863464 0.504410i \(-0.831710\pi\)
−0.863464 + 0.504410i \(0.831710\pi\)
\(164\) 10.3574 0.808780
\(165\) 0 0
\(166\) 1.76707 0.137151
\(167\) 1.64969 0.127657 0.0638285 0.997961i \(-0.479669\pi\)
0.0638285 + 0.997961i \(0.479669\pi\)
\(168\) 0 0
\(169\) −11.7408 −0.903136
\(170\) 6.88464 0.528027
\(171\) 0 0
\(172\) 10.0350 0.765160
\(173\) −15.6391 −1.18902 −0.594510 0.804088i \(-0.702654\pi\)
−0.594510 + 0.804088i \(0.702654\pi\)
\(174\) 0 0
\(175\) −0.690676 −0.0522102
\(176\) 3.49396 0.263367
\(177\) 0 0
\(178\) −7.71461 −0.578234
\(179\) −22.4432 −1.67748 −0.838742 0.544530i \(-0.816708\pi\)
−0.838742 + 0.544530i \(0.816708\pi\)
\(180\) 0 0
\(181\) −24.4845 −1.81992 −0.909959 0.414699i \(-0.863887\pi\)
−0.909959 + 0.414699i \(0.863887\pi\)
\(182\) 1.50161 0.111307
\(183\) 0 0
\(184\) −2.21836 −0.163540
\(185\) 15.8465 1.16506
\(186\) 0 0
\(187\) 8.88515 0.649747
\(188\) −1.69202 −0.123403
\(189\) 0 0
\(190\) −1.20486 −0.0874094
\(191\) −8.40818 −0.608394 −0.304197 0.952609i \(-0.598388\pi\)
−0.304197 + 0.952609i \(0.598388\pi\)
\(192\) 0 0
\(193\) −11.2466 −0.809546 −0.404773 0.914417i \(-0.632650\pi\)
−0.404773 + 0.914417i \(0.632650\pi\)
\(194\) −4.23748 −0.304233
\(195\) 0 0
\(196\) 2.00644 0.143317
\(197\) −9.27816 −0.661042 −0.330521 0.943799i \(-0.607224\pi\)
−0.330521 + 0.943799i \(0.607224\pi\)
\(198\) 0 0
\(199\) 11.4479 0.811519 0.405760 0.913980i \(-0.367007\pi\)
0.405760 + 0.913980i \(0.367007\pi\)
\(200\) −0.586887 −0.0414992
\(201\) 0 0
\(202\) −8.01443 −0.563894
\(203\) 1.46924 0.103120
\(204\) 0 0
\(205\) 13.2899 0.928206
\(206\) −2.49516 −0.173846
\(207\) 0 0
\(208\) −2.52146 −0.174832
\(209\) −1.55496 −0.107559
\(210\) 0 0
\(211\) −14.7941 −1.01846 −0.509232 0.860629i \(-0.670071\pi\)
−0.509232 + 0.860629i \(0.670071\pi\)
\(212\) 14.4581 0.992983
\(213\) 0 0
\(214\) −0.256469 −0.0175319
\(215\) 12.8761 0.878144
\(216\) 0 0
\(217\) −5.78466 −0.392689
\(218\) −0.187026 −0.0126670
\(219\) 0 0
\(220\) 5.71214 0.385113
\(221\) −6.41209 −0.431324
\(222\) 0 0
\(223\) −10.4239 −0.698036 −0.349018 0.937116i \(-0.613485\pi\)
−0.349018 + 0.937116i \(0.613485\pi\)
\(224\) −12.8877 −0.861098
\(225\) 0 0
\(226\) −1.62660 −0.108200
\(227\) −25.1619 −1.67005 −0.835026 0.550211i \(-0.814547\pi\)
−0.835026 + 0.550211i \(0.814547\pi\)
\(228\) 0 0
\(229\) 5.28237 0.349069 0.174535 0.984651i \(-0.444158\pi\)
0.174535 + 0.984651i \(0.444158\pi\)
\(230\) −1.30449 −0.0860158
\(231\) 0 0
\(232\) 1.24845 0.0819649
\(233\) −0.574578 −0.0376418 −0.0188209 0.999823i \(-0.505991\pi\)
−0.0188209 + 0.999823i \(0.505991\pi\)
\(234\) 0 0
\(235\) −2.17107 −0.141625
\(236\) −10.9389 −0.712059
\(237\) 0 0
\(238\) −7.64629 −0.495635
\(239\) −14.1140 −0.912959 −0.456480 0.889734i \(-0.650890\pi\)
−0.456480 + 0.889734i \(0.650890\pi\)
\(240\) 0 0
\(241\) −1.65454 −0.106578 −0.0532892 0.998579i \(-0.516971\pi\)
−0.0532892 + 0.998579i \(0.516971\pi\)
\(242\) 4.76271 0.306159
\(243\) 0 0
\(244\) −10.0467 −0.643173
\(245\) 2.57451 0.164479
\(246\) 0 0
\(247\) 1.12216 0.0714011
\(248\) −4.91539 −0.312128
\(249\) 0 0
\(250\) −6.36939 −0.402836
\(251\) 21.8280 1.37777 0.688886 0.724870i \(-0.258100\pi\)
0.688886 + 0.724870i \(0.258100\pi\)
\(252\) 0 0
\(253\) −1.68355 −0.105844
\(254\) 9.13271 0.573037
\(255\) 0 0
\(256\) −3.34721 −0.209200
\(257\) 0.602480 0.0375817 0.0187908 0.999823i \(-0.494018\pi\)
0.0187908 + 0.999823i \(0.494018\pi\)
\(258\) 0 0
\(259\) −17.5996 −1.09359
\(260\) −4.12225 −0.255651
\(261\) 0 0
\(262\) −7.49306 −0.462923
\(263\) 21.9560 1.35387 0.676933 0.736045i \(-0.263309\pi\)
0.676933 + 0.736045i \(0.263309\pi\)
\(264\) 0 0
\(265\) 18.5515 1.13961
\(266\) 1.33815 0.0820472
\(267\) 0 0
\(268\) 1.50093 0.0916836
\(269\) −7.92100 −0.482952 −0.241476 0.970407i \(-0.577631\pi\)
−0.241476 + 0.970407i \(0.577631\pi\)
\(270\) 0 0
\(271\) −10.2602 −0.623262 −0.311631 0.950203i \(-0.600875\pi\)
−0.311631 + 0.950203i \(0.600875\pi\)
\(272\) 12.8394 0.778504
\(273\) 0 0
\(274\) 3.95869 0.239153
\(275\) −0.445399 −0.0268586
\(276\) 0 0
\(277\) 4.50962 0.270957 0.135478 0.990780i \(-0.456743\pi\)
0.135478 + 0.990780i \(0.456743\pi\)
\(278\) −5.01887 −0.301012
\(279\) 0 0
\(280\) −10.7261 −0.641009
\(281\) 17.8458 1.06459 0.532295 0.846559i \(-0.321330\pi\)
0.532295 + 0.846559i \(0.321330\pi\)
\(282\) 0 0
\(283\) 18.1685 1.08001 0.540004 0.841662i \(-0.318423\pi\)
0.540004 + 0.841662i \(0.318423\pi\)
\(284\) −21.4420 −1.27235
\(285\) 0 0
\(286\) 0.968350 0.0572597
\(287\) −14.7601 −0.871264
\(288\) 0 0
\(289\) 15.6507 0.920629
\(290\) 0.734146 0.0431105
\(291\) 0 0
\(292\) 15.0929 0.883248
\(293\) −8.91987 −0.521105 −0.260552 0.965460i \(-0.583905\pi\)
−0.260552 + 0.965460i \(0.583905\pi\)
\(294\) 0 0
\(295\) −14.0359 −0.817202
\(296\) −14.9549 −0.869235
\(297\) 0 0
\(298\) −3.99296 −0.231306
\(299\) 1.21496 0.0702628
\(300\) 0 0
\(301\) −14.3006 −0.824273
\(302\) 7.50717 0.431989
\(303\) 0 0
\(304\) −2.24698 −0.128873
\(305\) −12.8911 −0.738145
\(306\) 0 0
\(307\) 20.2611 1.15636 0.578180 0.815909i \(-0.303763\pi\)
0.578180 + 0.815909i \(0.303763\pi\)
\(308\) −6.34408 −0.361488
\(309\) 0 0
\(310\) −2.89047 −0.164168
\(311\) −18.6975 −1.06024 −0.530119 0.847923i \(-0.677853\pi\)
−0.530119 + 0.847923i \(0.677853\pi\)
\(312\) 0 0
\(313\) −13.9380 −0.787820 −0.393910 0.919149i \(-0.628878\pi\)
−0.393910 + 0.919149i \(0.628878\pi\)
\(314\) 3.34451 0.188741
\(315\) 0 0
\(316\) −25.0117 −1.40702
\(317\) −3.17367 −0.178251 −0.0891256 0.996020i \(-0.528407\pi\)
−0.0891256 + 0.996020i \(0.528407\pi\)
\(318\) 0 0
\(319\) 0.947472 0.0530482
\(320\) 3.31699 0.185425
\(321\) 0 0
\(322\) 1.44881 0.0807391
\(323\) −5.71408 −0.317940
\(324\) 0 0
\(325\) 0.321428 0.0178296
\(326\) 12.2357 0.677672
\(327\) 0 0
\(328\) −12.5421 −0.692523
\(329\) 2.41126 0.132937
\(330\) 0 0
\(331\) 29.9695 1.64727 0.823636 0.567119i \(-0.191942\pi\)
0.823636 + 0.567119i \(0.191942\pi\)
\(332\) 5.38764 0.295685
\(333\) 0 0
\(334\) −0.915510 −0.0500945
\(335\) 1.92587 0.105222
\(336\) 0 0
\(337\) 13.2518 0.721874 0.360937 0.932590i \(-0.382457\pi\)
0.360937 + 0.932590i \(0.382457\pi\)
\(338\) 6.51563 0.354404
\(339\) 0 0
\(340\) 20.9907 1.13838
\(341\) −3.73038 −0.202011
\(342\) 0 0
\(343\) −19.7382 −1.06576
\(344\) −12.1516 −0.655172
\(345\) 0 0
\(346\) 8.67905 0.466589
\(347\) 18.8008 1.00928 0.504640 0.863330i \(-0.331625\pi\)
0.504640 + 0.863330i \(0.331625\pi\)
\(348\) 0 0
\(349\) 5.86566 0.313981 0.156991 0.987600i \(-0.449821\pi\)
0.156991 + 0.987600i \(0.449821\pi\)
\(350\) 0.383296 0.0204880
\(351\) 0 0
\(352\) −8.31096 −0.442976
\(353\) 16.1878 0.861587 0.430794 0.902450i \(-0.358234\pi\)
0.430794 + 0.902450i \(0.358234\pi\)
\(354\) 0 0
\(355\) −27.5128 −1.46023
\(356\) −23.5212 −1.24662
\(357\) 0 0
\(358\) 12.4550 0.658269
\(359\) −7.84290 −0.413933 −0.206966 0.978348i \(-0.566359\pi\)
−0.206966 + 0.978348i \(0.566359\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 13.5879 0.714162
\(363\) 0 0
\(364\) 4.57829 0.239968
\(365\) 19.3661 1.01367
\(366\) 0 0
\(367\) 32.0247 1.67168 0.835838 0.548976i \(-0.184982\pi\)
0.835838 + 0.548976i \(0.184982\pi\)
\(368\) −2.43280 −0.126819
\(369\) 0 0
\(370\) −8.79414 −0.457186
\(371\) −20.6038 −1.06970
\(372\) 0 0
\(373\) −14.3490 −0.742961 −0.371481 0.928441i \(-0.621150\pi\)
−0.371481 + 0.928441i \(0.621150\pi\)
\(374\) −4.93089 −0.254970
\(375\) 0 0
\(376\) 2.04892 0.105665
\(377\) −0.683756 −0.0352152
\(378\) 0 0
\(379\) 14.3470 0.736956 0.368478 0.929637i \(-0.379879\pi\)
0.368478 + 0.929637i \(0.379879\pi\)
\(380\) −3.67350 −0.188447
\(381\) 0 0
\(382\) 4.66619 0.238743
\(383\) −18.0149 −0.920518 −0.460259 0.887785i \(-0.652243\pi\)
−0.460259 + 0.887785i \(0.652243\pi\)
\(384\) 0 0
\(385\) −8.14025 −0.414865
\(386\) 6.24138 0.317678
\(387\) 0 0
\(388\) −12.9197 −0.655900
\(389\) −5.50923 −0.279329 −0.139665 0.990199i \(-0.544602\pi\)
−0.139665 + 0.990199i \(0.544602\pi\)
\(390\) 0 0
\(391\) −6.18662 −0.312871
\(392\) −2.42965 −0.122716
\(393\) 0 0
\(394\) 5.14899 0.259402
\(395\) −32.0931 −1.61478
\(396\) 0 0
\(397\) −28.6033 −1.43556 −0.717779 0.696271i \(-0.754841\pi\)
−0.717779 + 0.696271i \(0.754841\pi\)
\(398\) −6.35310 −0.318452
\(399\) 0 0
\(400\) −0.643620 −0.0321810
\(401\) −31.5073 −1.57340 −0.786699 0.617336i \(-0.788212\pi\)
−0.786699 + 0.617336i \(0.788212\pi\)
\(402\) 0 0
\(403\) 2.69208 0.134102
\(404\) −24.4353 −1.21570
\(405\) 0 0
\(406\) −0.815365 −0.0404659
\(407\) −11.3495 −0.562575
\(408\) 0 0
\(409\) 18.8319 0.931175 0.465588 0.885002i \(-0.345843\pi\)
0.465588 + 0.885002i \(0.345843\pi\)
\(410\) −7.37533 −0.364241
\(411\) 0 0
\(412\) −7.60753 −0.374796
\(413\) 15.5887 0.767070
\(414\) 0 0
\(415\) 6.91301 0.339346
\(416\) 5.99772 0.294062
\(417\) 0 0
\(418\) 0.862937 0.0422076
\(419\) −13.6195 −0.665358 −0.332679 0.943040i \(-0.607953\pi\)
−0.332679 + 0.943040i \(0.607953\pi\)
\(420\) 0 0
\(421\) −12.4308 −0.605842 −0.302921 0.953016i \(-0.597962\pi\)
−0.302921 + 0.953016i \(0.597962\pi\)
\(422\) 8.21008 0.399660
\(423\) 0 0
\(424\) −17.5077 −0.850248
\(425\) −1.63673 −0.0793929
\(426\) 0 0
\(427\) 14.3173 0.692862
\(428\) −0.781954 −0.0377972
\(429\) 0 0
\(430\) −7.14571 −0.344597
\(431\) −19.5788 −0.943077 −0.471539 0.881845i \(-0.656301\pi\)
−0.471539 + 0.881845i \(0.656301\pi\)
\(432\) 0 0
\(433\) −11.6960 −0.562073 −0.281036 0.959697i \(-0.590678\pi\)
−0.281036 + 0.959697i \(0.590678\pi\)
\(434\) 3.21025 0.154097
\(435\) 0 0
\(436\) −0.570227 −0.0273089
\(437\) 1.08270 0.0517925
\(438\) 0 0
\(439\) 16.3673 0.781169 0.390585 0.920567i \(-0.372273\pi\)
0.390585 + 0.920567i \(0.372273\pi\)
\(440\) −6.91700 −0.329755
\(441\) 0 0
\(442\) 3.55844 0.169258
\(443\) −30.3496 −1.44195 −0.720977 0.692959i \(-0.756307\pi\)
−0.720977 + 0.692959i \(0.756307\pi\)
\(444\) 0 0
\(445\) −30.1806 −1.43070
\(446\) 5.78483 0.273920
\(447\) 0 0
\(448\) −3.68395 −0.174050
\(449\) −5.18878 −0.244874 −0.122437 0.992476i \(-0.539071\pi\)
−0.122437 + 0.992476i \(0.539071\pi\)
\(450\) 0 0
\(451\) −9.51843 −0.448205
\(452\) −4.95936 −0.233269
\(453\) 0 0
\(454\) 13.9638 0.655353
\(455\) 5.87452 0.275402
\(456\) 0 0
\(457\) −21.4738 −1.00450 −0.502252 0.864721i \(-0.667495\pi\)
−0.502252 + 0.864721i \(0.667495\pi\)
\(458\) −2.93150 −0.136980
\(459\) 0 0
\(460\) −3.97730 −0.185442
\(461\) 22.7990 1.06186 0.530928 0.847417i \(-0.321843\pi\)
0.530928 + 0.847417i \(0.321843\pi\)
\(462\) 0 0
\(463\) 26.0362 1.21001 0.605003 0.796223i \(-0.293172\pi\)
0.605003 + 0.796223i \(0.293172\pi\)
\(464\) 1.36914 0.0635606
\(465\) 0 0
\(466\) 0.318867 0.0147712
\(467\) −36.1054 −1.67076 −0.835379 0.549674i \(-0.814752\pi\)
−0.835379 + 0.549674i \(0.814752\pi\)
\(468\) 0 0
\(469\) −2.13893 −0.0987668
\(470\) 1.20486 0.0555758
\(471\) 0 0
\(472\) 13.2462 0.609704
\(473\) −9.22208 −0.424032
\(474\) 0 0
\(475\) 0.286438 0.0131427
\(476\) −23.3129 −1.06854
\(477\) 0 0
\(478\) 7.83268 0.358258
\(479\) 18.9055 0.863812 0.431906 0.901919i \(-0.357841\pi\)
0.431906 + 0.901919i \(0.357841\pi\)
\(480\) 0 0
\(481\) 8.19053 0.373456
\(482\) 0.918202 0.0418230
\(483\) 0 0
\(484\) 14.5211 0.660050
\(485\) −16.5776 −0.752751
\(486\) 0 0
\(487\) 41.9276 1.89992 0.949961 0.312367i \(-0.101122\pi\)
0.949961 + 0.312367i \(0.101122\pi\)
\(488\) 12.1658 0.550721
\(489\) 0 0
\(490\) −1.42874 −0.0645441
\(491\) −20.5547 −0.927621 −0.463811 0.885934i \(-0.653518\pi\)
−0.463811 + 0.885934i \(0.653518\pi\)
\(492\) 0 0
\(493\) 3.48172 0.156809
\(494\) −0.622750 −0.0280188
\(495\) 0 0
\(496\) −5.39055 −0.242043
\(497\) 30.5565 1.37065
\(498\) 0 0
\(499\) 10.0538 0.450068 0.225034 0.974351i \(-0.427751\pi\)
0.225034 + 0.974351i \(0.427751\pi\)
\(500\) −19.4197 −0.868478
\(501\) 0 0
\(502\) −12.1136 −0.540658
\(503\) 11.3391 0.505585 0.252792 0.967521i \(-0.418651\pi\)
0.252792 + 0.967521i \(0.418651\pi\)
\(504\) 0 0
\(505\) −31.3536 −1.39522
\(506\) 0.934300 0.0415347
\(507\) 0 0
\(508\) 27.8449 1.23542
\(509\) −28.2990 −1.25433 −0.627166 0.778885i \(-0.715785\pi\)
−0.627166 + 0.778885i \(0.715785\pi\)
\(510\) 0 0
\(511\) −21.5086 −0.951485
\(512\) −21.2174 −0.937687
\(513\) 0 0
\(514\) −0.334351 −0.0147476
\(515\) −9.76141 −0.430139
\(516\) 0 0
\(517\) 1.55496 0.0683870
\(518\) 9.76704 0.429139
\(519\) 0 0
\(520\) 4.99175 0.218903
\(521\) −4.81055 −0.210754 −0.105377 0.994432i \(-0.533605\pi\)
−0.105377 + 0.994432i \(0.533605\pi\)
\(522\) 0 0
\(523\) −26.5418 −1.16059 −0.580297 0.814405i \(-0.697063\pi\)
−0.580297 + 0.814405i \(0.697063\pi\)
\(524\) −22.8457 −0.998021
\(525\) 0 0
\(526\) −12.1847 −0.531277
\(527\) −13.7082 −0.597138
\(528\) 0 0
\(529\) −21.8278 −0.949033
\(530\) −10.2953 −0.447199
\(531\) 0 0
\(532\) 4.07990 0.176886
\(533\) 6.86910 0.297534
\(534\) 0 0
\(535\) −1.00334 −0.0433784
\(536\) −1.81751 −0.0785046
\(537\) 0 0
\(538\) 4.39582 0.189517
\(539\) −1.84390 −0.0794225
\(540\) 0 0
\(541\) −42.4110 −1.82339 −0.911697 0.410864i \(-0.865227\pi\)
−0.911697 + 0.410864i \(0.865227\pi\)
\(542\) 5.69398 0.244577
\(543\) 0 0
\(544\) −30.5407 −1.30942
\(545\) −0.731672 −0.0313414
\(546\) 0 0
\(547\) −8.60780 −0.368043 −0.184021 0.982922i \(-0.558912\pi\)
−0.184021 + 0.982922i \(0.558912\pi\)
\(548\) 12.0697 0.515593
\(549\) 0 0
\(550\) 0.247178 0.0105397
\(551\) −0.609323 −0.0259580
\(552\) 0 0
\(553\) 35.6436 1.51572
\(554\) −2.50265 −0.106327
\(555\) 0 0
\(556\) −15.3021 −0.648955
\(557\) 11.4448 0.484931 0.242466 0.970160i \(-0.422044\pi\)
0.242466 + 0.970160i \(0.422044\pi\)
\(558\) 0 0
\(559\) 6.65524 0.281487
\(560\) −11.7630 −0.497077
\(561\) 0 0
\(562\) −9.90367 −0.417761
\(563\) −20.6340 −0.869619 −0.434809 0.900523i \(-0.643184\pi\)
−0.434809 + 0.900523i \(0.643184\pi\)
\(564\) 0 0
\(565\) −6.36348 −0.267713
\(566\) −10.0828 −0.423811
\(567\) 0 0
\(568\) 25.9647 1.08946
\(569\) 30.9846 1.29894 0.649470 0.760387i \(-0.274991\pi\)
0.649470 + 0.760387i \(0.274991\pi\)
\(570\) 0 0
\(571\) −3.21829 −0.134681 −0.0673407 0.997730i \(-0.521451\pi\)
−0.0673407 + 0.997730i \(0.521451\pi\)
\(572\) 2.95242 0.123447
\(573\) 0 0
\(574\) 8.19126 0.341897
\(575\) 0.310126 0.0129331
\(576\) 0 0
\(577\) 7.38475 0.307431 0.153716 0.988115i \(-0.450876\pi\)
0.153716 + 0.988115i \(0.450876\pi\)
\(578\) −8.68548 −0.361268
\(579\) 0 0
\(580\) 2.23835 0.0929424
\(581\) −7.67780 −0.318529
\(582\) 0 0
\(583\) −13.2869 −0.550286
\(584\) −18.2765 −0.756286
\(585\) 0 0
\(586\) 4.95016 0.204489
\(587\) 29.9644 1.23676 0.618381 0.785878i \(-0.287789\pi\)
0.618381 + 0.785878i \(0.287789\pi\)
\(588\) 0 0
\(589\) 2.39902 0.0988499
\(590\) 7.78934 0.320682
\(591\) 0 0
\(592\) −16.4005 −0.674057
\(593\) 8.21408 0.337312 0.168656 0.985675i \(-0.446057\pi\)
0.168656 + 0.985675i \(0.446057\pi\)
\(594\) 0 0
\(595\) −29.9133 −1.22633
\(596\) −12.1742 −0.498675
\(597\) 0 0
\(598\) −0.674250 −0.0275722
\(599\) −7.14594 −0.291975 −0.145988 0.989286i \(-0.546636\pi\)
−0.145988 + 0.989286i \(0.546636\pi\)
\(600\) 0 0
\(601\) −27.0946 −1.10521 −0.552606 0.833443i \(-0.686366\pi\)
−0.552606 + 0.833443i \(0.686366\pi\)
\(602\) 7.93624 0.323457
\(603\) 0 0
\(604\) 22.8887 0.931330
\(605\) 18.6324 0.757514
\(606\) 0 0
\(607\) −34.9253 −1.41757 −0.708787 0.705423i \(-0.750757\pi\)
−0.708787 + 0.705423i \(0.750757\pi\)
\(608\) 5.34481 0.216761
\(609\) 0 0
\(610\) 7.15404 0.289659
\(611\) −1.12216 −0.0453976
\(612\) 0 0
\(613\) −11.6878 −0.472064 −0.236032 0.971745i \(-0.575847\pi\)
−0.236032 + 0.971745i \(0.575847\pi\)
\(614\) −11.2440 −0.453773
\(615\) 0 0
\(616\) 7.68223 0.309526
\(617\) −7.48690 −0.301411 −0.150706 0.988579i \(-0.548155\pi\)
−0.150706 + 0.988579i \(0.548155\pi\)
\(618\) 0 0
\(619\) −5.22620 −0.210059 −0.105029 0.994469i \(-0.533494\pi\)
−0.105029 + 0.994469i \(0.533494\pi\)
\(620\) −8.81281 −0.353931
\(621\) 0 0
\(622\) 10.3763 0.416053
\(623\) 33.5195 1.34293
\(624\) 0 0
\(625\) −23.4858 −0.939431
\(626\) 7.73498 0.309152
\(627\) 0 0
\(628\) 10.1971 0.406910
\(629\) −41.7066 −1.66295
\(630\) 0 0
\(631\) 23.0356 0.917033 0.458517 0.888686i \(-0.348381\pi\)
0.458517 + 0.888686i \(0.348381\pi\)
\(632\) 30.2874 1.20477
\(633\) 0 0
\(634\) 1.76125 0.0699484
\(635\) 35.7284 1.41784
\(636\) 0 0
\(637\) 1.33068 0.0527234
\(638\) −0.525807 −0.0208169
\(639\) 0 0
\(640\) −25.0488 −0.990139
\(641\) 4.73368 0.186969 0.0934845 0.995621i \(-0.470199\pi\)
0.0934845 + 0.995621i \(0.470199\pi\)
\(642\) 0 0
\(643\) −5.06886 −0.199896 −0.0999481 0.994993i \(-0.531868\pi\)
−0.0999481 + 0.994993i \(0.531868\pi\)
\(644\) 4.41731 0.174066
\(645\) 0 0
\(646\) 3.17107 0.124764
\(647\) −36.1324 −1.42051 −0.710256 0.703943i \(-0.751421\pi\)
−0.710256 + 0.703943i \(0.751421\pi\)
\(648\) 0 0
\(649\) 10.0527 0.394605
\(650\) −0.178379 −0.00699660
\(651\) 0 0
\(652\) 37.3056 1.46100
\(653\) −24.3079 −0.951240 −0.475620 0.879651i \(-0.657776\pi\)
−0.475620 + 0.879651i \(0.657776\pi\)
\(654\) 0 0
\(655\) −29.3139 −1.14539
\(656\) −13.7545 −0.537024
\(657\) 0 0
\(658\) −1.33815 −0.0521665
\(659\) −25.5620 −0.995756 −0.497878 0.867247i \(-0.665887\pi\)
−0.497878 + 0.867247i \(0.665887\pi\)
\(660\) 0 0
\(661\) −6.32231 −0.245909 −0.122955 0.992412i \(-0.539237\pi\)
−0.122955 + 0.992412i \(0.539237\pi\)
\(662\) −16.6318 −0.646414
\(663\) 0 0
\(664\) −6.52405 −0.253182
\(665\) 5.23503 0.203006
\(666\) 0 0
\(667\) −0.659713 −0.0255442
\(668\) −2.79131 −0.107999
\(669\) 0 0
\(670\) −1.06878 −0.0412906
\(671\) 9.23284 0.356430
\(672\) 0 0
\(673\) 45.6443 1.75946 0.879730 0.475474i \(-0.157723\pi\)
0.879730 + 0.475474i \(0.157723\pi\)
\(674\) −7.35422 −0.283274
\(675\) 0 0
\(676\) 19.8656 0.764063
\(677\) −36.7205 −1.41128 −0.705641 0.708570i \(-0.749341\pi\)
−0.705641 + 0.708570i \(0.749341\pi\)
\(678\) 0 0
\(679\) 18.4116 0.706573
\(680\) −25.4182 −0.974744
\(681\) 0 0
\(682\) 2.07020 0.0792722
\(683\) −12.3880 −0.474015 −0.237007 0.971508i \(-0.576167\pi\)
−0.237007 + 0.971508i \(0.576167\pi\)
\(684\) 0 0
\(685\) 15.4870 0.591726
\(686\) 10.9538 0.418220
\(687\) 0 0
\(688\) −13.3263 −0.508060
\(689\) 9.58865 0.365298
\(690\) 0 0
\(691\) −23.1252 −0.879722 −0.439861 0.898066i \(-0.644972\pi\)
−0.439861 + 0.898066i \(0.644972\pi\)
\(692\) 26.4617 1.00592
\(693\) 0 0
\(694\) −10.4337 −0.396056
\(695\) −19.6345 −0.744781
\(696\) 0 0
\(697\) −34.9778 −1.32488
\(698\) −3.25519 −0.123211
\(699\) 0 0
\(700\) 1.16864 0.0441704
\(701\) 16.2845 0.615056 0.307528 0.951539i \(-0.400498\pi\)
0.307528 + 0.951539i \(0.400498\pi\)
\(702\) 0 0
\(703\) 7.29892 0.275284
\(704\) −2.37568 −0.0895369
\(705\) 0 0
\(706\) −8.98353 −0.338100
\(707\) 34.8223 1.30963
\(708\) 0 0
\(709\) 26.8960 1.01010 0.505049 0.863090i \(-0.331474\pi\)
0.505049 + 0.863090i \(0.331474\pi\)
\(710\) 15.2684 0.573014
\(711\) 0 0
\(712\) 28.4825 1.06743
\(713\) 2.59742 0.0972740
\(714\) 0 0
\(715\) 3.78832 0.141675
\(716\) 37.9744 1.41917
\(717\) 0 0
\(718\) 4.35248 0.162433
\(719\) 30.4185 1.13442 0.567210 0.823573i \(-0.308023\pi\)
0.567210 + 0.823573i \(0.308023\pi\)
\(720\) 0 0
\(721\) 10.8413 0.403752
\(722\) −0.554958 −0.0206534
\(723\) 0 0
\(724\) 41.4283 1.53967
\(725\) −0.174533 −0.00648200
\(726\) 0 0
\(727\) −38.6270 −1.43260 −0.716298 0.697795i \(-0.754165\pi\)
−0.716298 + 0.697795i \(0.754165\pi\)
\(728\) −5.54399 −0.205474
\(729\) 0 0
\(730\) −10.7474 −0.397779
\(731\) −33.8888 −1.25342
\(732\) 0 0
\(733\) −34.8178 −1.28602 −0.643012 0.765856i \(-0.722315\pi\)
−0.643012 + 0.765856i \(0.722315\pi\)
\(734\) −17.7724 −0.655990
\(735\) 0 0
\(736\) 5.78682 0.213305
\(737\) −1.37934 −0.0508087
\(738\) 0 0
\(739\) 17.0830 0.628407 0.314204 0.949356i \(-0.398263\pi\)
0.314204 + 0.949356i \(0.398263\pi\)
\(740\) −26.8126 −0.985651
\(741\) 0 0
\(742\) 11.4343 0.419765
\(743\) 40.4746 1.48487 0.742435 0.669918i \(-0.233671\pi\)
0.742435 + 0.669918i \(0.233671\pi\)
\(744\) 0 0
\(745\) −15.6210 −0.572310
\(746\) 7.96308 0.291549
\(747\) 0 0
\(748\) −15.0339 −0.549693
\(749\) 1.11434 0.0407173
\(750\) 0 0
\(751\) −48.0783 −1.75440 −0.877200 0.480124i \(-0.840592\pi\)
−0.877200 + 0.480124i \(0.840592\pi\)
\(752\) 2.24698 0.0819389
\(753\) 0 0
\(754\) 0.379456 0.0138190
\(755\) 29.3691 1.06885
\(756\) 0 0
\(757\) −36.0978 −1.31200 −0.655999 0.754762i \(-0.727753\pi\)
−0.655999 + 0.754762i \(0.727753\pi\)
\(758\) −7.96198 −0.289192
\(759\) 0 0
\(760\) 4.44835 0.161359
\(761\) −0.468230 −0.0169733 −0.00848667 0.999964i \(-0.502701\pi\)
−0.00848667 + 0.999964i \(0.502701\pi\)
\(762\) 0 0
\(763\) 0.812617 0.0294187
\(764\) 14.2268 0.514708
\(765\) 0 0
\(766\) 9.99751 0.361225
\(767\) −7.25470 −0.261952
\(768\) 0 0
\(769\) −51.5572 −1.85920 −0.929600 0.368569i \(-0.879848\pi\)
−0.929600 + 0.368569i \(0.879848\pi\)
\(770\) 4.51750 0.162799
\(771\) 0 0
\(772\) 19.0294 0.684885
\(773\) −24.2651 −0.872756 −0.436378 0.899763i \(-0.643739\pi\)
−0.436378 + 0.899763i \(0.643739\pi\)
\(774\) 0 0
\(775\) 0.687170 0.0246839
\(776\) 15.6449 0.561618
\(777\) 0 0
\(778\) 3.05739 0.109613
\(779\) 6.12134 0.219320
\(780\) 0 0
\(781\) 19.7051 0.705103
\(782\) 3.43332 0.122775
\(783\) 0 0
\(784\) −2.66452 −0.0951613
\(785\) 13.0842 0.466994
\(786\) 0 0
\(787\) 9.85684 0.351359 0.175679 0.984447i \(-0.443788\pi\)
0.175679 + 0.984447i \(0.443788\pi\)
\(788\) 15.6988 0.559248
\(789\) 0 0
\(790\) 17.8103 0.633663
\(791\) 7.06747 0.251290
\(792\) 0 0
\(793\) −6.66301 −0.236610
\(794\) 15.8736 0.563334
\(795\) 0 0
\(796\) −19.3701 −0.686554
\(797\) 40.8891 1.44837 0.724184 0.689607i \(-0.242217\pi\)
0.724184 + 0.689607i \(0.242217\pi\)
\(798\) 0 0
\(799\) 5.71408 0.202150
\(800\) 1.53096 0.0541275
\(801\) 0 0
\(802\) 17.4852 0.617425
\(803\) −13.8703 −0.489474
\(804\) 0 0
\(805\) 5.66795 0.199769
\(806\) −1.49399 −0.0526235
\(807\) 0 0
\(808\) 29.5895 1.04095
\(809\) −14.1624 −0.497923 −0.248961 0.968513i \(-0.580089\pi\)
−0.248961 + 0.968513i \(0.580089\pi\)
\(810\) 0 0
\(811\) 8.74292 0.307006 0.153503 0.988148i \(-0.450945\pi\)
0.153503 + 0.988148i \(0.450945\pi\)
\(812\) −2.48598 −0.0872408
\(813\) 0 0
\(814\) 6.29851 0.220763
\(815\) 47.8677 1.67673
\(816\) 0 0
\(817\) 5.93076 0.207491
\(818\) −10.4509 −0.365407
\(819\) 0 0
\(820\) −22.4868 −0.785272
\(821\) 7.74011 0.270132 0.135066 0.990837i \(-0.456875\pi\)
0.135066 + 0.990837i \(0.456875\pi\)
\(822\) 0 0
\(823\) 14.5574 0.507437 0.253719 0.967278i \(-0.418346\pi\)
0.253719 + 0.967278i \(0.418346\pi\)
\(824\) 9.21218 0.320922
\(825\) 0 0
\(826\) −8.65108 −0.301010
\(827\) −33.0581 −1.14954 −0.574772 0.818314i \(-0.694909\pi\)
−0.574772 + 0.818314i \(0.694909\pi\)
\(828\) 0 0
\(829\) 20.1795 0.700863 0.350431 0.936588i \(-0.386035\pi\)
0.350431 + 0.936588i \(0.386035\pi\)
\(830\) −3.83643 −0.133165
\(831\) 0 0
\(832\) 1.71444 0.0594376
\(833\) −6.77588 −0.234770
\(834\) 0 0
\(835\) −3.58160 −0.123947
\(836\) 2.63102 0.0909958
\(837\) 0 0
\(838\) 7.55827 0.261096
\(839\) −10.2578 −0.354137 −0.177068 0.984199i \(-0.556661\pi\)
−0.177068 + 0.984199i \(0.556661\pi\)
\(840\) 0 0
\(841\) −28.6287 −0.987197
\(842\) 6.89860 0.237741
\(843\) 0 0
\(844\) 25.0319 0.861632
\(845\) 25.4901 0.876885
\(846\) 0 0
\(847\) −20.6937 −0.711044
\(848\) −19.2001 −0.659334
\(849\) 0 0
\(850\) 0.908315 0.0311550
\(851\) 7.90253 0.270895
\(852\) 0 0
\(853\) −28.1505 −0.963853 −0.481927 0.876212i \(-0.660063\pi\)
−0.481927 + 0.876212i \(0.660063\pi\)
\(854\) −7.94550 −0.271889
\(855\) 0 0
\(856\) 0.946891 0.0323641
\(857\) −45.5987 −1.55762 −0.778812 0.627258i \(-0.784177\pi\)
−0.778812 + 0.627258i \(0.784177\pi\)
\(858\) 0 0
\(859\) −9.82703 −0.335294 −0.167647 0.985847i \(-0.553617\pi\)
−0.167647 + 0.985847i \(0.553617\pi\)
\(860\) −21.7867 −0.742919
\(861\) 0 0
\(862\) 10.8654 0.370077
\(863\) −38.1966 −1.30023 −0.650114 0.759836i \(-0.725279\pi\)
−0.650114 + 0.759836i \(0.725279\pi\)
\(864\) 0 0
\(865\) 33.9537 1.15446
\(866\) 6.49078 0.220566
\(867\) 0 0
\(868\) 9.78777 0.332219
\(869\) 22.9856 0.779733
\(870\) 0 0
\(871\) 0.995421 0.0337286
\(872\) 0.690504 0.0233834
\(873\) 0 0
\(874\) −0.600852 −0.0203241
\(875\) 27.6746 0.935573
\(876\) 0 0
\(877\) −14.1234 −0.476914 −0.238457 0.971153i \(-0.576642\pi\)
−0.238457 + 0.971153i \(0.576642\pi\)
\(878\) −9.08317 −0.306542
\(879\) 0 0
\(880\) −7.58564 −0.255712
\(881\) −12.4137 −0.418230 −0.209115 0.977891i \(-0.567058\pi\)
−0.209115 + 0.977891i \(0.567058\pi\)
\(882\) 0 0
\(883\) 11.8420 0.398514 0.199257 0.979947i \(-0.436147\pi\)
0.199257 + 0.979947i \(0.436147\pi\)
\(884\) 10.8494 0.364905
\(885\) 0 0
\(886\) 16.8428 0.565844
\(887\) 33.4723 1.12389 0.561945 0.827174i \(-0.310053\pi\)
0.561945 + 0.827174i \(0.310053\pi\)
\(888\) 0 0
\(889\) −39.6811 −1.33086
\(890\) 16.7490 0.561427
\(891\) 0 0
\(892\) 17.6375 0.590546
\(893\) −1.00000 −0.0334637
\(894\) 0 0
\(895\) 48.7258 1.62873
\(896\) 27.8199 0.929398
\(897\) 0 0
\(898\) 2.87956 0.0960921
\(899\) −1.46178 −0.0487530
\(900\) 0 0
\(901\) −48.8259 −1.62663
\(902\) 5.28233 0.175882
\(903\) 0 0
\(904\) 6.00543 0.199738
\(905\) 53.1576 1.76702
\(906\) 0 0
\(907\) 7.90462 0.262469 0.131234 0.991351i \(-0.458106\pi\)
0.131234 + 0.991351i \(0.458106\pi\)
\(908\) 42.5744 1.41288
\(909\) 0 0
\(910\) −3.26011 −0.108072
\(911\) 58.3641 1.93369 0.966845 0.255366i \(-0.0821959\pi\)
0.966845 + 0.255366i \(0.0821959\pi\)
\(912\) 0 0
\(913\) −4.95121 −0.163861
\(914\) 11.9171 0.394182
\(915\) 0 0
\(916\) −8.93789 −0.295316
\(917\) 32.5569 1.07512
\(918\) 0 0
\(919\) 38.3322 1.26446 0.632231 0.774780i \(-0.282139\pi\)
0.632231 + 0.774780i \(0.282139\pi\)
\(920\) 4.81622 0.158786
\(921\) 0 0
\(922\) −12.6525 −0.416688
\(923\) −14.2204 −0.468071
\(924\) 0 0
\(925\) 2.09069 0.0687413
\(926\) −14.4490 −0.474824
\(927\) 0 0
\(928\) −3.25672 −0.106907
\(929\) −14.0042 −0.459462 −0.229731 0.973254i \(-0.573785\pi\)
−0.229731 + 0.973254i \(0.573785\pi\)
\(930\) 0 0
\(931\) 1.18582 0.0388637
\(932\) 0.972198 0.0318454
\(933\) 0 0
\(934\) 20.0370 0.655630
\(935\) −19.2903 −0.630861
\(936\) 0 0
\(937\) 10.0677 0.328897 0.164448 0.986386i \(-0.447416\pi\)
0.164448 + 0.986386i \(0.447416\pi\)
\(938\) 1.18702 0.0387575
\(939\) 0 0
\(940\) 3.67350 0.119816
\(941\) −25.4924 −0.831028 −0.415514 0.909587i \(-0.636398\pi\)
−0.415514 + 0.909587i \(0.636398\pi\)
\(942\) 0 0
\(943\) 6.62757 0.215823
\(944\) 14.5266 0.472802
\(945\) 0 0
\(946\) 5.11787 0.166396
\(947\) −10.5429 −0.342597 −0.171299 0.985219i \(-0.554796\pi\)
−0.171299 + 0.985219i \(0.554796\pi\)
\(948\) 0 0
\(949\) 10.0097 0.324929
\(950\) −0.158961 −0.00515737
\(951\) 0 0
\(952\) 28.2303 0.914948
\(953\) 18.5540 0.601023 0.300511 0.953778i \(-0.402843\pi\)
0.300511 + 0.953778i \(0.402843\pi\)
\(954\) 0 0
\(955\) 18.2548 0.590711
\(956\) 23.8812 0.772373
\(957\) 0 0
\(958\) −10.4917 −0.338973
\(959\) −17.2003 −0.555426
\(960\) 0 0
\(961\) −25.2447 −0.814345
\(962\) −4.54540 −0.146550
\(963\) 0 0
\(964\) 2.79952 0.0901665
\(965\) 24.4171 0.786016
\(966\) 0 0
\(967\) 32.2043 1.03562 0.517809 0.855496i \(-0.326748\pi\)
0.517809 + 0.855496i \(0.326748\pi\)
\(968\) −17.5840 −0.565172
\(969\) 0 0
\(970\) 9.19989 0.295391
\(971\) −31.1805 −1.00063 −0.500316 0.865843i \(-0.666783\pi\)
−0.500316 + 0.865843i \(0.666783\pi\)
\(972\) 0 0
\(973\) 21.8067 0.699091
\(974\) −23.2681 −0.745558
\(975\) 0 0
\(976\) 13.3418 0.427062
\(977\) −34.9687 −1.11875 −0.559374 0.828916i \(-0.688958\pi\)
−0.559374 + 0.828916i \(0.688958\pi\)
\(978\) 0 0
\(979\) 21.6159 0.690846
\(980\) −4.35612 −0.139151
\(981\) 0 0
\(982\) 11.4070 0.364012
\(983\) 46.5994 1.48629 0.743145 0.669131i \(-0.233333\pi\)
0.743145 + 0.669131i \(0.233333\pi\)
\(984\) 0 0
\(985\) 20.1436 0.641828
\(986\) −1.93221 −0.0615340
\(987\) 0 0
\(988\) −1.89871 −0.0604061
\(989\) 6.42122 0.204183
\(990\) 0 0
\(991\) −43.7372 −1.38936 −0.694679 0.719320i \(-0.744453\pi\)
−0.694679 + 0.719320i \(0.744453\pi\)
\(992\) 12.8223 0.407109
\(993\) 0 0
\(994\) −16.9576 −0.537862
\(995\) −24.8542 −0.787932
\(996\) 0 0
\(997\) 56.7378 1.79691 0.898453 0.439070i \(-0.144692\pi\)
0.898453 + 0.439070i \(0.144692\pi\)
\(998\) −5.57941 −0.176613
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8037.2.a.i.1.3 6
3.2 odd 2 2679.2.a.i.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2679.2.a.i.1.4 6 3.2 odd 2
8037.2.a.i.1.3 6 1.1 even 1 trivial